Problem: Integrate. $ \int 2\csc^2(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $-2\cot(x)+C$ (Choice B) B $2\cot(x)+C$ (Choice C) C $-\dfrac{2}{3}\csc^3(x)+C$ (Choice D) D $\dfrac{2}{3}\csc^3(x)+C$
Solution: We need a function whose derivative is $2\csc^2(x)$. We know that the derivative of $\cot(x)$ is $-\csc^2(x)$, so let's start there: $\dfrac{d}{dx} \cot(x) = -\csc^2(x)$ Now let's multiply by $-2$ : $\dfrac{d}{dx} \left[ -2\cot(x) \right]= -2\dfrac{d}{dx} \cot(x) = 2\csc^2(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 2\csc^2(x)\,dx =-2 \cot(x)\, + C$ The answer: $-2 \cot(x)\, + C$